5.1 Hwk Answers due 29 Sept 15

To be done neatly in the front of your books
TOTAL 22 marks

1. The plots shows the curve $y=x^2+4x-1$

   

   1. Use the formula to solve the equation $x^2+4x-1=0$ giving your answers to 2 d.p. $$(x+2)^2 - 5 = 0 \\ x = -2 \pm \sqrt{5} \\ x=0.24 \text{ or } -4.24$$
   2. $x^2+4x-1=k$ has no solutions, where k is an integer. Use the plot above to find the maximum value of k $$k = -6$$

   [4 marks]

2. Complete the square $x^2+8x-4$ and use your answer to solve $$x^2+8x-4=0 \\ (x+4)^2 - 20 = 0 \\ x=-4 \pm \sqrt{20} \\ x=-4 \pm 2\sqrt{5}$$

   [3 marks]

3. Y varies indirectly as the square root of x. If y=7 when x=4, then:
   1. Find a formula for y in terms of x $$y=\frac{k}{\sqrt x} \\ k = 7 \times 2=14 \\ \therefore y=\frac{14}{\sqrt x}$$
   2. Find y when x=10 to 3 s.f. $$y = 14 \div \sqrt{10} = 4.43$$
   3. Find x when y=2.2 $$x=(\frac{14}{y})^2 = (\frac{14}{2.2})^2 = 40.5$$

   [4 marks]

4. Find the image of the point (-2, -3) under the transformation represented by M $$\begin{pmatrix} 4 & -3 \\ -5 & -7 \end{pmatrix} \begin{pmatrix} -2 \\ -3 \end{pmatrix} = \begin{pmatrix} 1 \\ 31 \end{pmatrix} \\ \text{Point is } (1,31)$$

   [2 marks]

5. Find the 2x2 matrix which represents a reflection in the x axis. $$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$

   [2 marks]

6. Write $2x^2+6x -3$ in the form $a(x+b)^2 +c$. $$2(x+\frac{3}{2})^2 - \frac{15}{2}$$

   [3 marks]

7. Draw up a table of values for $-2 \leq x \leq 3$ for the function $y=x^3 -2x^2+x-3$ and use it to sketch the curve neatly (and with appropriate scales). $$\begin{array}{|c|c|c|c|c|c|c|} \hline \ x & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline \ y & -21 & -7 & -3 & -3 & -1 & 9 \\ \hline \end{array}$$

   

   [4 marks]